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<li><a class="reference internal" href="#">Principal components analysis (PCA)</a></li>
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<p class="admonition-title">Note</p>
<p>Click <a class="reference internal" href="#sphx-glr-download-auto-examples-decomposition-plot-pca-3d-py"><span class="std std-ref">here</span></a> to download the full example code or to run this example in your browser via Binder</p>
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<div class="sphx-glr-example-title section" id="principal-components-analysis-pca">
<span id="sphx-glr-auto-examples-decomposition-plot-pca-3d-py"></span><h1>Principal components analysis (PCA)<a class="headerlink" href="#principal-components-analysis-pca" title="Permalink to this headline">¶</a></h1>
<p>These figures aid in illustrating how a point cloud
can be very flat in one direction–which is where PCA
comes in to choose a direction that is not flat.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="vm">__doc__</span><span class="p">)</span>

<span class="c1"># Authors: Gael Varoquaux</span>
<span class="c1">#          Jaques Grobler</span>
<span class="c1">#          Kevin Hughes</span>
<span class="c1"># License: BSD 3 clause</span>

<span class="kn">from</span> <span class="nn">sklearn.decomposition</span> <span class="kn">import</span> <span class="n">PCA</span>

<span class="kn">from</span> <span class="nn">mpl_toolkits.mplot3d</span> <span class="kn">import</span> <span class="n">Axes3D</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="kn">from</span> <span class="nn">scipy</span> <span class="kn">import</span> <span class="n">stats</span>


<span class="c1"># #############################################################################</span>
<span class="c1"># Create the data</span>

<span class="n">e</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">4</span><span class="p">)</span>


<span class="k">def</span> <span class="nf">pdf</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
    <span class="k">return</span> <span class="mf">0.5</span> <span class="o">*</span> <span class="p">(</span><span class="n">stats</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">scale</span><span class="o">=</span><span class="mf">0.25</span> <span class="o">/</span> <span class="n">e</span><span class="p">)</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">x</span><span class="p">)</span>
                  <span class="o">+</span> <span class="n">stats</span><span class="o">.</span><span class="n">norm</span><span class="p">(</span><span class="n">scale</span><span class="o">=</span><span class="mi">4</span> <span class="o">/</span> <span class="n">e</span><span class="p">)</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>

<span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">scale</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="p">(</span><span class="mi">30000</span><span class="p">))</span>
<span class="n">x</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">scale</span><span class="o">=</span><span class="mf">0.5</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="p">(</span><span class="mi">30000</span><span class="p">))</span>
<span class="n">z</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="n">scale</span><span class="o">=</span><span class="mf">0.1</span><span class="p">,</span> <span class="n">size</span><span class="o">=</span><span class="nb">len</span><span class="p">(</span><span class="n">x</span><span class="p">))</span>

<span class="n">density</span> <span class="o">=</span> <span class="n">pdf</span><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">*</span> <span class="n">pdf</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>
<span class="n">pdf_z</span> <span class="o">=</span> <span class="n">pdf</span><span class="p">(</span><span class="mi">5</span> <span class="o">*</span> <span class="n">z</span><span class="p">)</span>

<span class="n">density</span> <span class="o">*=</span> <span class="n">pdf_z</span>

<span class="n">a</span> <span class="o">=</span> <span class="n">x</span> <span class="o">+</span> <span class="n">y</span>
<span class="n">b</span> <span class="o">=</span> <span class="mi">2</span> <span class="o">*</span> <span class="n">y</span>
<span class="n">c</span> <span class="o">=</span> <span class="n">a</span> <span class="o">-</span> <span class="n">b</span> <span class="o">+</span> <span class="n">z</span>

<span class="n">norm</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">a</span><span class="o">.</span><span class="n">var</span><span class="p">()</span> <span class="o">+</span> <span class="n">b</span><span class="o">.</span><span class="n">var</span><span class="p">())</span>
<span class="n">a</span> <span class="o">/=</span> <span class="n">norm</span>
<span class="n">b</span> <span class="o">/=</span> <span class="n">norm</span>


<span class="c1"># #############################################################################</span>
<span class="c1"># Plot the figures</span>
<span class="k">def</span> <span class="nf">plot_figs</span><span class="p">(</span><span class="n">fig_num</span><span class="p">,</span> <span class="n">elev</span><span class="p">,</span> <span class="n">azim</span><span class="p">):</span>
    <span class="n">fig</span> <span class="o">=</span> <span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">fig_num</span><span class="p">,</span> <span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">clf</span><span class="p">()</span>
    <span class="n">ax</span> <span class="o">=</span> <span class="n">Axes3D</span><span class="p">(</span><span class="n">fig</span><span class="p">,</span> <span class="n">rect</span><span class="o">=</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">0</span><span class="p">,</span> <span class="o">.</span><span class="mi">95</span><span class="p">,</span> <span class="mi">1</span><span class="p">],</span> <span class="n">elev</span><span class="o">=</span><span class="n">elev</span><span class="p">,</span> <span class="n">azim</span><span class="o">=</span><span class="n">azim</span><span class="p">)</span>

    <span class="n">ax</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">a</span><span class="p">[::</span><span class="mi">10</span><span class="p">],</span> <span class="n">b</span><span class="p">[::</span><span class="mi">10</span><span class="p">],</span> <span class="n">c</span><span class="p">[::</span><span class="mi">10</span><span class="p">],</span> <span class="n">c</span><span class="o">=</span><span class="n">density</span><span class="p">[::</span><span class="mi">10</span><span class="p">],</span> <span class="n">marker</span><span class="o">=</span><span class="s1">&#39;+&#39;</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=.</span><span class="mi">4</span><span class="p">)</span>
    <span class="n">Y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">c_</span><span class="p">[</span><span class="n">a</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">]</span>

    <span class="c1"># Using SciPy&#39;s SVD, this would be:</span>
    <span class="c1"># _, pca_score, V = scipy.linalg.svd(Y, full_matrices=False)</span>

    <span class="n">pca</span> <span class="o">=</span> <span class="n">PCA</span><span class="p">(</span><span class="n">n_components</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
    <span class="n">pca</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">Y</span><span class="p">)</span>
    <span class="n">pca_score</span> <span class="o">=</span> <span class="n">pca</span><span class="o">.</span><span class="n">explained_variance_ratio_</span>
    <span class="n">V</span> <span class="o">=</span> <span class="n">pca</span><span class="o">.</span><span class="n">components_</span>

    <span class="n">x_pca_axis</span><span class="p">,</span> <span class="n">y_pca_axis</span><span class="p">,</span> <span class="n">z_pca_axis</span> <span class="o">=</span> <span class="mi">3</span> <span class="o">*</span> <span class="n">V</span><span class="o">.</span><span class="n">T</span>
    <span class="n">x_pca_plane</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="n">x_pca_axis</span><span class="p">[:</span><span class="mi">2</span><span class="p">],</span> <span class="o">-</span> <span class="n">x_pca_axis</span><span class="p">[</span><span class="mi">1</span><span class="p">::</span><span class="o">-</span><span class="mi">1</span><span class="p">]]</span>
    <span class="n">y_pca_plane</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="n">y_pca_axis</span><span class="p">[:</span><span class="mi">2</span><span class="p">],</span> <span class="o">-</span> <span class="n">y_pca_axis</span><span class="p">[</span><span class="mi">1</span><span class="p">::</span><span class="o">-</span><span class="mi">1</span><span class="p">]]</span>
    <span class="n">z_pca_plane</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">r_</span><span class="p">[</span><span class="n">z_pca_axis</span><span class="p">[:</span><span class="mi">2</span><span class="p">],</span> <span class="o">-</span> <span class="n">z_pca_axis</span><span class="p">[</span><span class="mi">1</span><span class="p">::</span><span class="o">-</span><span class="mi">1</span><span class="p">]]</span>
    <span class="n">x_pca_plane</span><span class="o">.</span><span class="n">shape</span> <span class="o">=</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
    <span class="n">y_pca_plane</span><span class="o">.</span><span class="n">shape</span> <span class="o">=</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
    <span class="n">z_pca_plane</span><span class="o">.</span><span class="n">shape</span> <span class="o">=</span> <span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">)</span>
    <span class="n">ax</span><span class="o">.</span><span class="n">plot_surface</span><span class="p">(</span><span class="n">x_pca_plane</span><span class="p">,</span> <span class="n">y_pca_plane</span><span class="p">,</span> <span class="n">z_pca_plane</span><span class="p">)</span>
    <span class="n">ax</span><span class="o">.</span><span class="n">w_xaxis</span><span class="o">.</span><span class="n">set_ticklabels</span><span class="p">([])</span>
    <span class="n">ax</span><span class="o">.</span><span class="n">w_yaxis</span><span class="o">.</span><span class="n">set_ticklabels</span><span class="p">([])</span>
    <span class="n">ax</span><span class="o">.</span><span class="n">w_zaxis</span><span class="o">.</span><span class="n">set_ticklabels</span><span class="p">([])</span>


<span class="n">elev</span> <span class="o">=</span> <span class="o">-</span><span class="mi">40</span>
<span class="n">azim</span> <span class="o">=</span> <span class="o">-</span><span class="mi">80</span>
<span class="n">plot_figs</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">elev</span><span class="p">,</span> <span class="n">azim</span><span class="p">)</span>

<span class="n">elev</span> <span class="o">=</span> <span class="mi">30</span>
<span class="n">azim</span> <span class="o">=</span> <span class="mi">20</span>
<span class="n">plot_figs</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">elev</span><span class="p">,</span> <span class="n">azim</span><span class="p">)</span>

<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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